Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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Prove for criterion that two curve families are orthogonal on a surface in 3D

Let $E, F, G$ be the coefficients of the first fundamental form of a regular surface $R = R(u, v).$ Let $f(u, v) = c$ and $g(u, v) = d$ be two families of regular curves defined in the ...
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30 views

Change of basis formula proof

So I know that this involves using the chain rule, but is the following attempt at a proof correct. Let $M$ be an $n$-dimensional manifold and let $(U,\phi)$ and $(V,\psi)$ be two overlapping ...
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2answers
55 views

What is a good reference that connects calculus with differential geometry?

It seems that most texts on differential geometry books tend to take a quantum leap from calculus without refering the latter. Differentials suddenly becomes forms, functions suddenly becomes ...
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3answers
72 views

What is the volume and surface area of the 1-Sphere?

I am reading a post on here that mentioned something about the 1-sphere. I know that a 2-sphere is a circle, and 3-sphere is a volume, but what is this 1-sphere and how do you calculate the volume and ...
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2answers
20 views

Existence of a countable basis in the definition of a manifold and uncountable bases.

In the definition of a manifold $M$ of dimension $n$ in An Introduction to Differentiable Manifolds and Riemannian Geometry by William M. Boothby (page 6), the third criterion is $M$ has a ...
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1answer
48 views

What is a pullback in simple calculus context?

The definition of a pullback provided by my text is quite accessible Let $\phi : M \to N$, $f:N \to \mathbb{R}$, then $f\circ \phi: M \to \mathbb{R}$, where $\phi^*f = f\circ\phi$ and $\phi^*$ is ...
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1answer
72 views

Next book in learning Differential Geometry

I have just finished the book "Manfredo P. do Carmo - Differential Geometry of Curves and Surfaces". My aim is to reach to graduate level to do research, but articles are not only too advanced to ...
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2answers
78 views

What is the modern approach to tensors?

I have recently started studying tensors a bit from the index notation point of view -- I understand contractions and the metric tensor and such well enough. However I've been told that this approach ...
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18 views

Showing that the rank of the complex projective space is 1

I was assigned the task of calculating the rank of the complex projective space $\mathbb C P^n=SU(n)/S(U(1)\times U(n-1))$ and am not sure how best to approach that task. (looking in the ...
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2answers
49 views

Counting singularities

It is well known that a smooth vector field on a 2-sphere must vanish twice. What is the general technique for counting singularities of a smooth map between manifolds? For example, how many ...
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1answer
37 views

What is meant by “The Lie derivative commutes with contraction”?

This was stated recently in a GR course I am taking, and I found it also stated on Wikipedia (second paragraph). I simply don't know what is meant by this. For a vector $X$ and 1-form $\eta$, I would ...
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49 views

Is a polyhedron an affine manifold?

I was reading the definition of an affine manifold (https://www.wikiwand.com/en/Affine_manifold) and was wondering if a polyhedron is an affine manifold. Could you also provide any hints to the proof ...
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26 views

Vector Bundle Structure on $\sqcup_{p\in M}\mathcal L(T_pM, T_{f(p)}N)$.

Let $f:M\to N$ be a smooth map between smooth manifolds. Is there a natural way to give a smooth vector bundle structure to $\bigsqcup_{p\in M} \mathcal L(T_pM, T_{f(p)}N)$. where $\mathcal L(V, ...
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1answer
29 views

Concept of Partial Derivatives in the Context of Smooth Manifolds

The concept of a partial derivative is fundamental when studying multivariable calculus. I was wondering if there is a standard definition of partial derivatives in the context of smooth manifolds. I ...
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0answers
37 views

Pullback map distributes over wedge product (proof)

To prove that the pullback map distributes with the wedge product is it first best to prove that it distributes over the tensor product and then use the relation $$dx^{\mu_{1}}\wedge\cdots\wedge ...
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16 views

The Weierstrass-Enneper representation, the Gauss map

Lemma: Let $x:S\to\mathbb{R}^3$ be a conformal minimal immersion of a Riemann surface. The 1-forms $f_k=(x_{k,u}-ix_{k,v})dz$ satisfy: $$ \sum_kf_k^2=0\qquad (1)\qquad \&\qquad ...
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3answers
50 views

Example of orthogonal parametrization of a surface

I recently came to know about the orthogonal parametrization of a surface, for which $F={\bf X_u}\cdot{\bf X_v}=0$ and $E={\bf X_u}\cdot{\bf X_u}=G={\bf X_v}\cdot{\bf X_v}$. Here, $(E,F,G)$ denote the ...
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25 views

Normal coordinates

I was wondering if this is a legitimate way to define the induced basis of the tangent space in normal coordinates. So the exponential map is a diffemorphism $exp:U \subset T_pM \rightarrow V \subset ...
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3answers
476 views

Why is this map called a fold?

Consider the map $\varphi : \mathbb R^2 \to \mathbb R^2$ defined by $(x,y) \mapsto (x,y^2)$. Apparently this map is called a fold as the $(x,y)$-plane is folded over and creased along the axis ...
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16 views

Relation between parallel transport and Jacobi field II

Before I asked a question here: Relation between parallel vector field along a geodesic and Jacobi field along that same geodesic The current question is related, and actually arise from numerical ...
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2answers
48 views

What interpretation of the Lie braket is this?

I was reading exercise 1.96 of Gadea's Analysis and Algebra on Differentiable Manifolds. I don't know what definition of Lie braket the author used, but I'm confused, as far as I know ...
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Need some help understanding this exercise about injective plane curve

Let $\gamma (t) = (x(t), y(t))$ be a smooth regular plane curve $\gamma: I \to \mathbb R^2$ where $I$ is some open interval. Now consider the following exercise: Let $\varphi (u,v) = (x(u), v + ...
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1answer
113 views

Cancellation in topological product

I was wondering whether $M\times \mathbb{R}$ is homeomorphic to $N\times \mathbb{R}$ implies $M$ is homeomorphic to $N$, where let us say $M,N$ are smooth manifolds. (They are certainly homotopy ...
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40 views

Almost independent vectors- Where do they live on a manifold [on hold]

I am new to this thing. I am having the next question : Almost independent vectors- Where do they live on a manifold? In a manifold with larger dimmension? Tnks!So don't be tuff with me cause I am ...
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1answer
53 views

Preimage of singular points of smooth map between manifolds

Given a smooth ($C^{\infty}$) map $\phi: V \rightarrow SU(n)$ where $V$ is a (finite dim, real) vector space (of potentially very large dimension) and $SU(n)$ is the special unitary Lie group, what ...
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17 views

Hypersphere central angle

For a sphere, the relationship between steradian of a patch on the surface, and the central angle of the cone subtending that patch, is given by ...
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55 views

Does naturality for characteristic classes imply the classifying space is universal for them?

Let $G$ be a Lie group, $\mathfrak g$ its Lie algebra, $K$ its maximal compact subgroup. To every flat $G$-bundle $P$ over a smooth manifold $M$ I can associate a homomorphism $w_P: H^*(\mathfrak g, ...
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1answer
40 views

Angle form, 1-form, proof verification.

Check that the $1$-form $d\,\text{arg}$ in $\mathbb{R}^2 - \{0\}$ is just the form$${{-y}\over{x^2 + y^2}}\,dx + {{x}\over{x^2 + y^2}}\,dy.$$ My solution is as follows. Observe that we can ...
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2answers
145 views
+50

Prove that the sphere is the only closed surface in $\mathbb{R}^3$ that minimizes the surface area to volume ratio.

It is well known that a sphere minimizes the surface area to volume ratio since it reaches equality in the Isoperimetric Inequality. I'm trying to prove that no other closed surface has this property. ...
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1answer
19 views

Geodesic equation applied to halfplane model

I have learned some things regarding connections and geodesic. And I want to apply this knowledge to the exercise: show that the vertical lines in the halfplane model are geodesics. The metric is ...
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33 views

Stokes' theorem: Induced orientation on the boundary of a manifold

The Question Let $K = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \geq 1\}$, where $K$ is oriented via the canonical volume form on $\mathbb{R}^3$: $dx \wedge dy \wedge dz$. Let $\mathbb{S}^2$ be ...
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1answer
31 views

Normal coordinates and the metric tensor

I was wondering whether the metric tensor in normal coordinates can be expressed somehow in terms of the exponential map. Cause I just don't see how the metric in normal coordinates is actually ...
2
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1answer
6 views

Compound map in manifolds

In the description of a manifold, we often start with the mathematical definition that $M=\cup M_i$ and if $m\in M_i \subset M$, where m is a point on the manifold, then it is mapped by a one-to-one ...
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34 views

Proving that the pullback map commutes with the exterior derivative

I'm trying to prove that the pullback map $\phi^{\ast}$ induced by a map $\phi:M\rightarrow N$ commutes with the exterior derivative. Here is my attempt so far: Let $\omega\;\in\Omega^{r}(N)$ and let ...
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1answer
54 views

Questions about torsion of a curve in $\mathbb{R}^3$ and analogues of torsion in higher dimensions

Suppose we have a curve $\alpha(s) : I \to \mathbb{R}^3$ parametrized by arc-length that has nowhere-vanishing second derivative, so that we are able to define the torsion $\tau(s)$ for every $s \in ...
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2answers
46 views

How to prove that in a Kähler manifold without boundary $\Omega \wedge \cdots \wedge \Omega$ is closed but not exact?

Let $M$ be a compact Kähler manifold without boundary. How can I show that the volume form $$ \Omega^{m} = \Omega \wedge \cdots \wedge \Omega $$ where we have the wedge of $m$ $\Omega$s is ...
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0answers
20 views

Homogeneous metric on a homogeneous space $G/K$ - is this the same as a $G$ - invariant metric?

I have trouble putting down the notion of a homogeneous Riemannian metric. Suppose we are given a Riemannian manifold $(M,g)$ on which a compact Lie group $G$ acts transitively by isometries (this ...
2
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1answer
50 views

Divisors of differentials.

Let $\mathbb{C}$ be the base field. Suppose $C \subset \mathbb{P}_2$ is a nonsingular projective curve of degree $d > 3$. Must it be the case that $D \in \text{Div}(C)$ is the divisor of a ...
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1answer
28 views

Why is a surface of revolution injective?

Let $f:U \rightarrow \mathbb{R}$ and $g:U \rightarrow \mathbb{R}$ be smooth functions where $U \subset \mathbb{R}$ is an open set such that $f(x) > 0$ and $f'(x)^2 + g'(x)^2 = 1$ for any $x$ in ...
16
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1answer
96 views

Are there spaces that 'look the same' at every point, but are not homogeneous?

A metric space is homogeneous if for any two points there is a global isometry that maps one into the other. It is locally homogeneous if any two points have isometric neighborhoods, i.e. the space ...
3
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1answer
43 views

Does map induced by rotation preserve the volume form?

Let $A: \mathbb{R}^n \to \mathbb{R}^n$ be a rotation. My question is, does the map of $S^{n-1}$ onto $S^{n-1}$ induced by $A$ necessarily preserve the volume form?
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0answers
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G-P Exercise 4.8.2, proof verification.

Let $\gamma$ be a smooth closed curve in $\mathbb{R}^2 - \{0\}$ and $\omega$ any closed $1$-form on $\mathbb{R}^2 - \{0\}$. Prove that$$\oint_\gamma \omega = W(\gamma, 0) \int_{S^1} \omega,$$where ...
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1answer
25 views

Normal vector in curvilinear coordinates

Is it true that the normal vector, or, $\ddot{\mathbf r}$ always vanishes for: a helix in cylindrical coordinates a loxodrome in spherical coordinates a torus knot in toroidal coordinates When ...
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4answers
75 views

Why do disks on planes grow more quickly with radius than disks on spheres?

In the book, Mr. Tompkins in Wonderland, there is written something like this: On a sphere the area within a given radius grows more slowly with the radius than on a plane. Could you explain ...
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maximal linear subspaces contained in the cone over the Clifford torus.

Forgot: this is about Find a subspace of $\mathbb{R}^4$ for which $x^T*A*x$ = 0 I was a little surprised to find that, in the cone $x^2 + y^2 = z^2 + w^2$ in $\mathbb R^4,$ there are infinitely many ...
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37 views

How to compute the unit outer normal at the point in a curve?

Given a smooth closed curve $f(x,y)=0$, How to compute the unit outer normal at each point $(x_{0},y_{0})$ in the curve?
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1answer
291 views

Riemann tensor symmetries

The Riemann tensor has its component expression: ...
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1answer
33 views

Jacobi field along every geodesic?

I stumbled over the question: Given a manifold $M$. Can there exist a vector field that is a Jacobi field along every geodesic? Now, the answer is apparently: Yes, because the $0$ vector field does ...
2
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1answer
43 views

Need help understanding a relation between the fundamental forms

The book I am reading briefly mentions this relation between the fundamental forms but gives no explanation of how they got it. Take the following as the Weingarten Map/Shape Operator where $\nu$ is ...
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1answer
63 views

Can a $1d$ space never be curved?

I was wondering about this: Wikipedia article I refer to (here I refer to the first part: metric) This wikipedia article claims that this hyperbolic space model has constant curvature $-1.$ I believe ...